一类带调和势的非线性Schrödinger方程的解

研究了一类带调和势的Schrödinger方程的解，运用能量守衡定律和质量守衡定律以及利用矢量分析的知识，引入积分不等式和解微分不等式的方法，得到了初值满足一定条件的柯西问题的解会在有限的时间里发生爆破的结论。由于所讨论方程更具有一般性，从而推广了已有的结论，所得到结论也可以对能量和质量的集中现象作进一步解释。
关 键 词 调和势; 非线性Schrödinger方程; 爆破; 能量守衡

Abstract In the paper, a class of Schrödinger equation with harmonic Potential, which concerns Bose-Einstin condensates, is investigated. Now that the equation of Bose-einstin condensates describes lot of phenomena, that we research it has special significance. With the help of using energy conservative law and quality conservative law as well as knowledge of vector analysis, integral and differential inequality, we prove that the solution to the Cauchy’s problem will blow up in finite time in case initial value satisfy with some conditions. As the equation in the paper is more general, we have got extensive conclusion, by means of which we may deeply understand the aggregative phenomena on energy and quality.
Key words nonlinear Schrödinger equation; harmonic potential; blow up; energy